Method for evaluating radiation hydrodynamic force of concentric cylindrical structure

Abstract

The present invention relates to the technical field of coastal engineering, and disclosed is a method for evaluating a radiation hydrodynamic force for a concentric cylindrical structure, comprising the following steps: S1, coordinate system establishment: establishing a cylindrical coordinate system orθz and a Cartesian coordinate system oxyz by taking the intersection of the center of a concentric cylindrical structure and a still water surface as the origin, where the z-axis is vertically upward; S2, flow field division: dividing the entire flow field into an outer domain and an inner domain; S3, perforated boundary condition assumption; S4, determining radiation velocity potentials of the inner domain and the outer domain by means of governing equations and boundary conditions; and S5, by means of the expressions of velocity potential functions of the inner and outer domains, determining first-order radiation forces, added masses and damping coefficients generated by the vibration of an inner cylinder and an outer cylinder. The method takes into account both the effects of seismic and radiation hydrodynamic forces, provides accurate structural response prediction, fills the technological gap in the field of seismic analysis of concentric cylindrical structures as breakwater units, and provides a reliable basis for engineering design.

Description

Radiation hydrodynamic evaluation method for concentric cylindrical structures Technical Field

[0001] The invention belongs to the technical field of coastal engineering, and in particular relates to a radiation hydrodynamic evaluation method for a concentric cylindrical structure. Background Art

[0002] Breakwaters, as important coastal engineering structures, play a crucial role along coastlines or riverbanks, slowing down waves or river flows and protecting nearby areas from erosion or flooding. They are typically constructed from large stones, concrete, and other materials and can take various forms, such as straight or curved, to effectively absorb and reflect wave energy, thereby protecting shoreline facilities and land.

[0003] Specifically, the design of a concentric double-cylinder breakwater utilizes a marine structural unit consisting of an inner and outer cylinder, creating a bottom-seated structure with an annular gap between the two cylinders. Such a structure, arranged along the coastline, forms a breakwater. However, under the influence of natural disasters such as earthquakes, such breakwaters can be subject to forced oscillations, inducing wave disturbances that affect the stability and safety of the structure. Therefore, accurately assessing the structure's radiative hydrodynamic response is crucial. First, this can provide reasonable seismic input data during the design phase to ensure that the structure maintains good engineering performance in the event of a disaster, protecting nearby land and facilities. Second, accurately assessing the radiative hydrodynamic response can provide engineers with a basis for improving and optimizing the structure to ensure stability under various natural disaster conditions.

[0004] In summary, radiation hydrodynamic assessment of concentric double-cylinder breakwaters not only provides critical data for the design phase but also serves as a foundation for ensuring their seismic performance and engineering safety. This assessment is crucial for ensuring the effective function of breakwaters in coastal engineering projects while also contributing to the safety and stability of nearby areas. Therefore, it is necessary to develop an effective radiation hydrodynamic assessment method for concentric double-cylinder structures.

[0005] Summary of the Invention

[0006] In response to the problems existing in the prior art, the present invention provides an evaluation method for accurately predicting the radiation hydrodynamics of concentric cylindrical structures, which is difficult to solve with traditional methods.

[0007] The present invention is achieved by providing a method for evaluating the radiation hydrodynamics of a concentric cylindrical structure, which is characterized by comprising the following steps:

[0008] S1. Establish a coordinate system:

[0009] With the intersection of the center of the concentric cylindrical structure and the still water surface as the origin, establish the cylindrical coordinate system orθz and the Cartesian coordinate system oxyz, where the z axis is vertically upward;

[0010] S2. Divide the flow field into the outer and inner domains:

[0011] The entire flow field is divided into an outer domain (a2<r, 0≤θ≤2π, -h≤z≤0) and an inner domain (a1<r≤a2, 0≤θ≤2π, -h≤z≤0), which are represented by Ω0 and Ω1 respectively. In the framework of linear potential flow theory, assuming that the fluid is uniform, irrotational, inviscid and incompressible, the radiation velocity potential Φ that satisfies the Laplace equation can be used. kR (r, θ, z, t) is used to describe the fluid motion, k = 0 and 1 represent the outer domain and inner domain respectively. Extracting the time factor of the velocity potential, we can get: In the above formula, Re represents the real part of the complex number, φ kR represents the spatial factor of the radiation velocity potential, i represents the imaginary unit, ω represents the circular frequency of the wave motion, and t represents time.

[0012] Where h represents the water depth, a1 represents the radius of the inner cylinder, and a2 represents the radius of the outer cylinder;

[0013] S3. Opening boundary condition assumptions:

[0014] Set the boundary condition expression of the porous outer cylinder:

[0015] In the above formula, φ kR (k = 0, 1) represents the radiation velocity potential of the flow domain, where k = 0 represents the outer domain and k = 1 represents the inner domain; i represents the imaginary unit; the superscripts + and - represent the outer and inner surfaces of the outer cylinder, respectively; σ represents the porosity effect parameter;

[0016] δ j The following definitions are given:

[0017] S4. Determine the radiation velocity potential of the inner and outer domains using the following formula:

[0018] Radiation velocity potential within the structure The expression is:

[0019] Radiation velocity potential outside the structure The expression is:

[0020] In the above formula, Re represents the real part; i represents the imaginary unit; t represents time; ω represents the circular frequency of the wave;

[0021] ζ jrepresents the motion amplitude of the structure in the jth mode, where j = 1 represents the longitudinal mode and j = 2 represents the transverse mode;

[0022] represents the velocity potential generated when the inner cylinder oscillates at unit velocity in the j direction, and is defined as the normalized radiation velocity potential of the inner domain;

[0023] represents the velocity potential generated when the outer cylinder oscillates at unit velocity in the j direction, and is defined as the normalized radiation velocity potential of the outer domain;

[0024] S5. Determine the first-order radiation force, added mass, and damping coefficient in the i direction generated by the vibration of the inner and outer cylinders in mode j according to the following expressions, where i = 1 indicates the positive x-axis direction and i = 2 indicates the positive y-axis direction:

[0025] The first-order radiation force in the i direction generated by the vibration of the inner cylinder in mode j The expression:

[0026] The additional mass in the i direction generated by the vibration of the inner cylinder in mode j expression:

[0027] The damping coefficient of the inner cylinder in the j mode in the i direction expression:

[0028] Where ω represents the circular frequency of the wave; ρ represents the density of water; Re represents the real part of the complex variable; Im represents the imaginary part of the complex variable; the subscript I represents the inner cylinder; S I Represents the outer side of the inner cylinder; ζ j represents the motion amplitude of the structure in the jth mode, where j = 1 represents the longitudinal mode and j = 2 represents the transverse mode; n i is the unit normal vector component, here it is agreed

[0029] The first-order radiation force in the i direction generated by the vibration of the outer cylinder in mode j expression:

[0030] The additional mass in the i direction generated by the vibration of the outer cylinder in mode j expression:

[0031] The damping coefficient of the outer cylinder in the j mode vibration in the i direction expression:

[0032] In equations (28), (29), and (30), ω represents the circular frequency of the wave; ρ represents the density of water; Re represents the real part of the complex variable; Im represents the imaginary part of the complex variable; the subscript O represents the outer cylinder; S O Indicates the outer side of the outer cylinder; ζ j represents the motion amplitude of the structure in the jth mode, where j = 1 represents the longitudinal mode and j = 2 represents the transverse mode; n i is the unit normal vector component, here it is agreed

[0033] In the above technical solution, preferably, the method for evaluating the radiation hydrodynamics of concentric cylindrical structures according to claim 1 is characterized in that: in the coordinate system established in step S1, the seabed plane is S B , the free surface is S F , the water depth is h; where j = 1 represents the longitudinal mode of the structure oscillating along the x direction, and j = 2 represents the transverse mode of the structure oscillating along the y direction.

[0034] In the above technical solution, preferably, σ is calculated by the following formula: σ = μ / (ρlω), (2)

[0035] In formula (2), ρ represents the density of water; μ represents the dynamic viscosity of water; l represents the porosity coefficient with length unit;

[0036] ω represents the wave circular frequency.

[0037] In the above technical solution, preferably, in step S4, the spatial factors of the velocity potential of the inner domain and the outer domain are first determined:

[0038] Assuming that the structure oscillates with a circular frequency of ω, the instantaneous displacement ζ of the structure is j (t) is expressed as: j (t) = Re[ζ j e -iωt ], (3)

[0039] In formula (3), Re represents the real part of the complex variable; i represents the imaginary unit; t represents time; ζ j Indicates the motion amplitude of the structure in the jth mode; the structural vibration velocity The expression is:

[0040] Radiation velocity potential within the structure The expression of the satisfied physical surface condition is:

[0041] In formula (5), n j is the unit normal vector component, here it is agreed

[0042] According to the radiation velocity potential within the structure The object surface condition expression that satisfies the inner domain radiation velocity potential is set to:

[0043] In formula (6), is the normalized radiation velocity potential of the inner domain, that is, the velocity potential generated when the inner cylinder oscillates at unit velocity in the j direction; the object surface condition expression of the inner cylinder is:

[0044] In the above technical solution, preferably, the outer domain normalized radiation velocity potential is derived The process is:

[0045] In the outer domain of the flow field, the normalized radiation velocity potential of the concentric cylindrical structure in the j direction is The following governing equations and boundary conditions are satisfied:

[0046] In formula (9), ν = ω 2 / g; ω represents the circular frequency of wave motion; g represents the acceleration of gravity; k0 represents the wave number of radiation wave; the external domain normalized radiation velocity potential that satisfies formula (9) is The expression is:

[0047] In formula (10), A mn Represents the unknown coefficient; H m represents the first-kind m-order Hankel function; K m represents the second-kind m-order modified Bessel function; where the outer domain radial characteristic function expression is:

[0048] The vertical characteristic function expression of the outer domain is:

[0049] k0 and k n (n=1,2,3,...) is the positive real root of the following dispersion relation:

[0050] Derivation of the normalized radiation velocity potential in the inner domain The process is:

[0051] In the inner domain of the flow field, the normalized radiation velocity potential of the concentric cylindrical structure in the j direction is The following governing equations and boundary conditions are satisfied:

[0052] in, n1 indicates that the structure oscillates along the x-axis; n2 indicates that the structure oscillates along the y-axis;

[0053] The normalized radiation velocity potential that satisfies the first three equations of (14) and the first equation of (1) is Expressed as:

[0054] In formula (15), C mn and D mn is the unknown coefficient;

[0055] Q m (k n r) and W m (k n r) is defined as follows:

[0056] J m and Y m denote the first and second kind m-order Bessel functions respectively;

[0057] I m and K m denote the first and second kind m-order modified Bessel functions respectively;

[0058] Establish the linear equations of the radiation velocity potential in the inner and outer domains:

[0059] The unknown coefficients in the radiation velocity potential of the inner and outer domains can be solved using the following transmission condition expressions:

[0060] Substitute equations (10) and (15) into the first equation of transmission condition (16), and according to Z n (z) Orthogonality on the interval [-h, 0] can be expressed as: A mn R m ′(k n a2)=C mn Q m ′(k n a2)+D mn W m ′(k n a2), (17)

[0061] in,

[0062] Substituting equations (10) and (15) into the second equation of the transmission condition (16), using the orthogonality of cos(mθ) and sin(mθ) in the interval [0,2π] and Z n The orthogonality of (z) on the interval [-h, 0] gives the expression:

[0063] where N n is the following expression:

[0064] δ m,j is the following expression:

[0065] Using the cylinder condition of the inner column, that is, substituting equations (10) and (15) into the fourth equation in (14), and again using the orthogonality of cos(mθ) and sin(mθ) on the interval [0,2π] and Z n The orthogonality of (z) on the interval [-h, 0] gives:

[0066] Solve equations (17)-(21) to find the unknown coefficient A mn 、C mn and D mn , and then the radiation velocity potential of the inner and outer domains can be determined. Through the velocity potential of the inner and outer radiation domains, the first-order radiation force, added mass and damping coefficient of the structure can be solved.

[0067] Advantages and effects of the present invention:

[0068] First, the radiation hydrodynamic analysis method for concentric cylindrical structures fills the research gap in the field of breakwater seismic analysis, provides a scientific and reliable technical basis for the design, construction and maintenance of such structures, ensures the stability and seismic performance of the structure, and has important engineering practice value.

[0069] Secondly, this method can comprehensively evaluate the stability and seismic performance of concentric cylindrical structures. By deeply studying the structural characteristics of stress and deformation, it provides more accurate reference data for engineering design and implementation, promoting the development of breakwater engineering.

[0070] Furthermore, the method is efficient and practical. Based on scientific analysis, it can effectively reduce research and implementation costs, improve engineering design efficiency, and contribute positively to technological advancements in coastal engineering and breakwaters.

[0071] In summary, this assessment method provides an unprecedented scientific basis for the design, construction, and maintenance of concentric cylindrical structures. It not only promotes the innovation and development of breakwater engineering, but also provides strong technical support for protecting the safety of coastlines and nearby areas. BRIEF DESCRIPTION OF THE DRAWINGS

[0072] Figures 1 and 2 are schematic diagrams of the coordinate system in the method of the present invention;

[0073] FIG3 is a schematic flow chart of the steps of the method of the present invention. DETAILED DESCRIPTION

[0074] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0075] To address the difficulty of accurately predicting the radiation hydrodynamic forces of concentric cylindrical structures using traditional methods, the present invention provides a method for assessing the radiation hydrodynamic forces of concentric cylindrical structures. This method comprehensively considers the effects of earthquakes and radiation hydrodynamic forces, providing accurate structural response predictions. This method fills a technical gap in the field of seismic analysis of concentric cylindrical structures used as breakwater units and provides a reliable basis for engineering design. To further illustrate the structure of the present invention, a detailed description is provided below with reference to the accompanying drawings:

[0076] Please refer to Figures 1 and 2, which are diagrams of the analytical model for radiation hydrodynamic analysis in the radiation hydrodynamic evaluation method for a concentric cylindrical structure proposed in the present invention. The radiation hydrodynamic analysis method for this structure, as shown in Figure 3, adopts the following steps:

[0077] S1. Establish a coordinate system.

[0078] Establish the coordinate system as shown in Figure 1. At the intersection of the center of the concentric cylindrical system and the still water surface, establish the cylindrical coordinate system orθz and the Cartesian coordinate system oxyz, where the z axis is vertically upward.

[0079] Assuming the seabed is flat and impermeable, S B Indicates that the free surface is S F The water depth is h. The inner concentric cylinder is rigid and impermeable, with a radius a1; the outer concentric cylinder has a rigid hole with a radius a2. The thickness of the outer hole is negligible relative to the wave wavelength. j (t) represents the instantaneous displacement of the structural oscillation, where j = 1 represents the longitudinal swell mode and j = 2 represents the transverse swell mode. Within the framework of radiation theory, the oscillation of a rigidly fixed structure is caused by earthquake action, i.e., the structure undergoes forced oscillations. Furthermore, the inner and outer columns are rigidly fixed, meaning there is no relative motion between them.

[0080] S2. Watershed Division

[0081] The flow field in the coordinate system is divided into the outer domain (a2<r, 0≤θ≤2π, -h≤z≤0) and the inner domain (a1<r≤a2, 0≤θ≤2π, -h≤z≤0), which are represented by Ω0 and Ω1 respectively. In the framework of linear potential flow theory, assuming that the fluid is uniform, irrotational, inviscid and incompressible, the radiation velocity potential Φ that satisfies the Laplace equation can be used. kR(r, θ, z, t) is used to describe the fluid motion, k = 0 and 1 represent the outer domain and inner domain respectively. Extracting the time factor of the velocity potential, we can get: In the above formula, Re represents the real part of the complex number, φ kR represents the spatial factor of the radiation velocity potential, i represents the imaginary unit, ω represents the circular frequency of the wave motion, and t represents time.

[0082] S3. Opening boundary condition assumptions

[0083] In this embodiment, the thickness of the porous outer column is negligible relative to the wave wavelength, and the porous surface is defined as fine pores. Therefore, the fluid flow on the porous surface can be described by linear Darcy's law. According to this law, the flow velocities inside and outside the porous surface are linearly proportional. Combined with the linear Bernoulli equation, the boundary conditions of the porous outer column can be expressed as follows: In the above formula, φ kR (k=0,1) represents the radiation velocity potential of the kth flow basin, i represents the imaginary unit, and the superscripts + and - of a2 represent the outer and inner surfaces of the outer column, respectively. j The following definitions are given:

[0084] σ represents the porous effect parameter of the outer column, which can be calculated by the following formula: σ = μ / (ρlω), (2)

[0085] Where ρ and μ represent the density and dynamic viscosity of water, l represents the porosity coefficient in units of length, and ω represents the wave circular frequency. When σ = 0, the surface is completely porous, equivalent to non-existent. As σ increases, the surface porosity decreases. When σ increases to infinity, the surface is completely non-porous.

[0086] S4. Determine the radiation velocity potential of the inner and outer domains.

[0087] First, determine the spatial factors of the velocity potential in the inner and outer domains:

[0088] Assuming that due to the action of the earthquake, the structure oscillates with a circular frequency of ω, then the instantaneous displacement ζ j (t) can be expressed as: j (t) = Re[ζ j e -iωt ], (3) In the above formula, Re represents the real part, i represents the imaginary unit, t represents time, ζ j represents the motion amplitude of the structure in the jth mode. For a cylindrical structure under earthquake, only the translational mode is considered. The subscripts j = 1 and 2 represent the longitudinal sway mode (oscillation along the x-direction) and the transverse sway mode (oscillation along the y-direction), respectively.

[0089] According to formula (3), the structural vibration velocity It can be expressed as:

[0090] Internal radiation velocity potential of the structure The physical surface conditions that are satisfied are:

[0091] where n j is the unit normal vector component, here it is agreed

[0092] According to the right side form of the object surface condition (5), the internal radiation velocity potential is set to:

[0093] in, It is called the normalized radiation velocity potential of the inner domain, which represents the velocity potential generated when the inner column oscillates at unit velocity in the j direction. At this time, the object surface condition of the inner column is transformed into:

[0094] Similarly, the external radiation velocity potential is set to: in, represents the normalized radiation velocity potential generated when the outer cylinder oscillates with unit velocity in the j direction.

[0095] The derivation process of the normalized radiation velocity potential of the above external domain is:

[0096] In the outer domain of the flow field, the normalized radiation velocity potential of the concentric cylindrical structure in the j direction is The following governing equations and boundary conditions are satisfied:

[0097] In the above formula, ν=ω 2 / g, ω represents the circular frequency of wave motion, g represents the acceleration of gravity, and k0 represents the wave number of the incident wave.

[0098] The outer domain normalized radiation velocity potential that satisfies equation (9) Expressed as:

[0099] Among them, the radial characteristic function of the outer domain is:

[0100] The vertical characteristic function of the outer domain is:

[0101] In formulas (10)-(12), A mn represents the unknown coefficient, H m represents the first-kind m-order Hankel function, K m represents the m-th order modified Bessel function of the second kind.

[0102] k0 and k n (n=1,2,3,...) are the positive real roots of the following dispersion relation:

[0103] Derive the normalized radiation velocity potential in the inner domain:

[0104] In the inner domain of the flow field, the normalized radiation velocity potential of the concentric cylindrical structure in the j direction is The following governing equations and boundary conditions are satisfied:

[0105] in, n1 indicates that the structure oscillates along the x-axis, i.e., the longitudinal sway mode; n2 indicates that the structure oscillates along the y-axis, i.e., the transverse sway mode.

[0106] The normalized radiation velocity potential that satisfies the first three equations of (14) Expressed as:

[0107] in,

[0108] In the above formula, C mn and D mn is the unknown coefficient, J m and Y m Represent the first and second m-order Bessel functions, I m and K m Represent the first and second m-order modified Bessel functions respectively. The vertical characteristic function Z n (z) is the same as the vertical characteristic function of the external domain, see formula (12).

[0109] Establish the linear equations of the radiation velocity potential in the inner and outer domains:

[0110] Substitute equations (10) and (15) into the first equation of transmission condition (16), and according to Z n (z) Orthogonality on the interval [-h, 0], we can get: A mn R m ′(k n a2)=C mn Q m ′(k n a2)+D mn W m ′(k n a2), (17)

[0111] in:

[0112] Substituting equations (10) and (15) into the second equation of the transmission condition (16), using the orthogonality of cos(mθ) and sin(mθ) in the interval [0,2π] and Z n (z) Orthogonality on the interval [-h, 0] gives:

[0113] where N n is the following expression:

[0114] δ m,j is the following expression:

[0115] Finally, using the cylinder condition of the inner cylinder, that is, substituting equations (10) and (15) into the fourth equation in (14), and again using the orthogonality of cos(mθ) and sin(mθ) on the interval [0,2π] and Z n (z) Orthogonality on the interval [-h, 0] gives:

[0116] Among them, δ m,j The definition of is given in equation (20).

[0117] By combining equations (17)-(21), we can solve the unknown coefficient A mn 、C mn and D mn , and then the radiation velocity potential of the inner and outer domains can be determined. The radiation force, added mass, and damping coefficient of the structure can be solved through the velocity potential of the inner and outer radiation domains.

[0118] S5. Derive hydrodynamic expressions.

[0119] After determining the radiation velocity potential, the first-order radiation force in the i direction generated by the vibration of the inner column in mode j can be expressed as:

[0120] Separate the inner column radiation force from the acceleration and velocity of the object's corresponding mode of motion:

[0121] Combining equations (22) and (23), we get:

[0122] In the above formula, and represents the additional mass and damping coefficient generated by the vibration of the inner column in the j-mode in the i-direction. Equations (23) and (24) show that the force acting on the inner column consists of two parts. One part is proportional to the acceleration of the structural oscillation, and the proportional coefficient of this part is the additional mass; the other part is proportional to the speed of the structural oscillation, and its proportional coefficient is called the damping coefficient. The generation of this part of the force is a phenomenon unique to the presence of a free surface in the fluid, and represents the wave-making resistance caused by the structural oscillation. After further sorting out Equation (24), we have:

[0123] After further arrangement, the calculation expressions of the added mass and damping coefficient of the inner column can be obtained as follows:

[0124] Similar to the inner column, the first-order radiation force, added mass, and damping coefficient in the i-direction generated by the vibration of the outer column in mode j can be expressed as:

[0125] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for evaluating the radiation hydrodynamics of a concentric cylindrical structure, characterized in that The following steps are involved: S1. Establish the coordinate system: Take the intersection of the center of the concentric cylindrical structure and the still water surface as the origin, establish the cylindrical coordinate system orθz and the Cartesian coordinate system oxyz, where the z axis is vertically upward; S2. Divide the flow basin into the outer domain and the inner domain: The entire flow field is divided into an outer domain (a2<r, 0≤θ≤2π, -h≤z≤0) and an inner domain (a1<r≤a2, 0≤θ≤2π, -h≤z≤0), which are represented by Ω0 and Ω1 respectively; where h represents the water depth, a1 represents the inner cylinder radius, and a2 represents the outer cylinder radius; S3. Opening boundary condition assumptions: Set the opening boundary condition expression of the porous outer cylinder: In the above formula, φ kR (k=0,1) represents the radiation velocity potential of the flow domain, where k=0 represents the outer domain and k=1 represents the inner domain; i represents an imaginary unit; the superscripts + and - in the formula represent the outer surface and inner surface of the outer cylinder, respectively; σ represents the porous effect parameter; δ j The following definitions are given: S4. Determine the radiation velocity potential of the inner and outer domains by the following formula: Radiation velocity potential within the structure The expression is: Radiation velocity potential outside the structure The expression is: In the above formula, Re represents the real part; i represents the imaginary unit; t represents time; ω represents the circular frequency of the wave; ζ j represents the motion amplitude of the structure in the jth mode, where j = 1 represents the longitudinal mode and j = 2 represents the transverse mode; represents the velocity potential generated when the inner cylinder oscillates at unit velocity in the j direction, and is defined as the normalized radiation velocity potential in the inner domain; represents the velocity potential generated when the outer cylinder oscillates at unit velocity in the j direction, and is defined as the normalized radiation velocity potential in the outer domain; S5. Determine the first-order radiation force, additional mass, and damping coefficient generated by the vibration of the inner and outer cylinders in mode j in the i direction according to the following expressions, where i = 1 indicates the positive direction of the x-axis and i = 2 indicates the positive direction of the y-axis: The first-order radiation force in the i direction generated by the vibration of the inner cylinder in mode j The expression is: The additional mass in the i direction generated by the vibration of the inner cylinder in mode j expression: The damping coefficient of the inner cylinder in the j mode in the i direction expression: Where ω represents the circular frequency of the wave; ρ represents the density of water; Re represents the real part of the complex variable; Im represents the imaginary part of the complex variable; the subscript I represents the inner cylinder; S I Represents the outer side of the inner cylinder; ζ j represents the motion amplitude of the structure in the jth mode, where j = 1 represents the longitudinal mode and j = 2 represents the transverse mode; n i is the unit normal vector component, and is agreed upon here The first-order radiation force in the i direction generated by the vibration of the outer cylinder in mode j expression: The additional mass in the i direction generated by the vibration of the outer cylinder in mode j expression: The damping coefficient of the outer cylinder in the j mode in the i direction expression: In equations (28), (29), and (30), ω represents the circular frequency of the wave; ρ represents the density of water; Re represents the real part of the complex variable; Im represents the imaginary part of the complex variable; the subscript O represents the outer cylinder; S O Represents the outer side of the outer cylinder; ζ j represents the motion amplitude of the structure in the jth mode, where j = 1 represents the longitudinal mode and j = 2 represents the transverse mode; n i is the unit normal vector component, and is agreed upon here 2. The method for evaluating radiation hydrodynamics of concentric cylindrical structures according to claim 1, characterized in that: In the coordinate system established in step S1, the seabed plane is S B , the free surface is S F , the water depth is h; where j = 1 represents the longitudinal mode of the structure oscillating along the x direction, and j = 2 represents the transverse mode of the structure oscillating along the y direction.

3. The method for evaluating radiation hydrodynamics of concentric cylindrical structures according to claim 2, characterized in that: σ is calculated by the following formula: σ = μ / (ρlω), (2) In formula (2), ρ represents the density of water; μ represents the dynamic viscosity of water; l represents the porosity coefficient with length unit; ω represents the wave circular frequency.

4. The method for evaluating radiation hydrodynamics of concentric cylindrical structures according to claim 3 is characterized in that: In step S4, the spatial factors of the inner and outer domain velocity potentials are first determined: Assuming that the structure oscillates with a circular frequency of ω, the instantaneous displacement of the structure is j (t) is expressed as: ζ j (t)(Re[ζ j e -iωt ], (3) In formula (3), Re represents the real part of the complex variable; i represents the imaginary unit; t represents time; ζ j Represents the motion amplitude of the structure in the jth mode; structural vibration velocity The expression is: Radiation velocity potential within the structure The expression of the satisfied physical surface condition is: In formula (5), n j is the unit normal vector component, and is agreed upon here According to the radiation velocity potential in the structure The satisfied object surface condition expression sets the expression of the internal radiation velocity potential to: In formula (6), is the normalized radiation velocity potential of the inner domain, that is, the velocity potential generated when the inner cylinder oscillates at unit velocity in the j direction; the object surface condition expression of the inner cylinder is:

5. The method for evaluating radiation hydrodynamics of concentric cylindrical structures according to claim 4, characterized in that: Derivation of the normalized radiation velocity potential in the outer domain The process is: In the outer domain of the flow field, the normalized radiation velocity potential of the concentric cylindrical structure in the j direction is The following control equations and boundary conditions are satisfied: In formula (9), ν = ω 2 / g; ω represents the circular frequency of wave motion; g represents the gravitational acceleration; k0 represents the radiation wave number; the external domain normalized radiation velocity potential that satisfies equation (9) The expression is: In formula (10), A mn Represents the unknown coefficient; H m represents the first kind m-order Hankel function; K m represents the second-kind m-order modified Bessel function; where the outer domain radial characteristic function expression is: The vertical characteristic function expression of the external domain is: k0 and k n (n=1,2,3,...) is the positive real root of the following dispersion relation: Derivation of the normalized radiation velocity potential in the inner domain The process is: In the inner domain of the flow field, the normalized radiation velocity potential of the concentric cylindrical structure in the j direction is The following governing equations and boundary conditions are satisfied: in, n1 indicates that the structure oscillates along the x-axis; n2 indicates that the structure oscillates along the y-axis; the normalized radiation velocity potential that satisfies the first three equations of (14) and the first equation of (1) It is expressed as: In formula (15), C mn and D mn is the unknown coefficient; Q m (k n r) and W m (k n r) is defined as follows: J m and Y m denote the first and second kind m-order Bessel functions respectively; I m and K m denote the first and second kind m-order modified Bessel functions respectively; Establish the linear equations of the radiation velocity potential in the inner and outer domains: The unknown coefficients in the radiation velocity potential of the inner and outer domains can be solved using the following transmission condition expression: Substitute equations (10) and (15) into the first equation of transmission condition (16), and according to Z n The orthogonality of (z) on the interval [-h, 0] gives the expression: A mn R m ′(k n a2)=C mn Q m ′(k n a2)+D mn W m ′(k n a2), (17) in, Substituting equations (10) and (15) into the second equation of the transmission condition (16), using the orthogonality of cos(mθ) and sin(mθ) on the interval [0,2π] and Z n (z) Orthogonality on the interval [-h, 0] gives the expression: Where N n is the following expression: δ m,j The expression is: Using the cylinder condition of the inner cylinder, that is, substituting equations (10) and (15) into the fourth equation in (14), again using the orthogonality of cos(mθ) and sin(mθ) on the interval [0,2π] and Z n The orthogonality of (z) on the interval [-h, 0] gives: Combine equations (17)-(21) to solve for the unknown coefficient A mn , C mn and D mn , and then the radiation velocity potential of the inner and outer domains can be determined. Through the velocity potential of the inner and outer radiation domains, the first-order radiation force, added mass and damping coefficient of the structure can be solved.

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